Finite Group Databasebeta

← #13143942

Subgroups — #13143942

Order 1728.

891 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 891 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88744C211yesmaximal, characteristic
2C2×#88527C211yesmaximal, characteristic, complemented
3C4⋊C4×#272C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
5#11720578C211yesmaximal, characteristic, complemented
6#11720549C211yesmaximal, characteristic, complemented
7#11720532C211yesmaximal, characteristic, complemented
8C3×#4771—34nomaximal
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C2×#6235—21no—
11C22×#1084—21no—
12#88744—22no—
13#88744C414yescomplemented
14C2×C4×#272—21no—
15C2×C4×#272C2211yescharacteristic
16C2×#6229C2211yescharacteristic
17C2×C8×H(3,𝔽3)C2211yescharacteristic
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19C4⋊#1084C2211yescharacteristic
20#88540—21no—
21C2×#6180C2211yescharacteristic
22#88527—22no—
23C2×C3×#237—34no—
24C2×C3×ΓL(1,𝔽25)—34no—
25C3×D3×C4⋊C4—34no—
26C32×#102D314yescomplemented
27C3×#832—34no—
28C3×#816—34no—
29C3×#810—34no—
30C23×H(3,𝔽3)—21no—
31C22×#272D411yescharacteristic
32D4×H(3,𝔽3)—21no—
33D4×H(3,𝔽3)C2×C412yescomplemented
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)C2311yescharacteristic
36#6235—24no—
37C2×#1084—21no—
38C2×#1084—24no—
39C2×#1084—21no—
40C2×#1084D411yescharacteristic
41C4×#272—42no—
42C4×#272C2×C412yes—
43C4×#272D412yes—
44#6229—21no—
45#6229C2×C412yescomplemented
46C8×H(3,𝔽3)—21no—
47#6180—21no—
48C3×#417—91no—
49#4771—34no—
50C2×C32×D4C2×D314yes—
51C2×C3×#67—64no—
52C22×C3×Dic3—64no—
53C3×#237—316no—
54C3×#237—68no—
55C2×C3×C4×D3—34no—
56C2×C3×C4×D3—64no—
57C2×C3×Dic6—34no—
58C2×C32×C8C2×D314yes—
59C32×C4⋊C4C2×D314yes—
60C3×C4⋊Dic3—34no—
61C3×Dic3⋊C4—64no—
62C2×C3×C3⋊C8—34no—
63C3×ΓL(1,𝔽25)—68no—
64C22×H(3,𝔽3)—22no—
65C22×H(3,𝔽3)—41no—
66C22×H(3,𝔽3)—41no—
67C22×H(3,𝔽3)C2×D411yescharacteristic
68C2×#272—21no—
69C2×#272—21no—
70C2×#272C22⋊C412yes—
71C4×H(3,𝔽3)—41no—
72C4×H(3,𝔽3)C22×C411yescharacteristic
73C4×H(3,𝔽3)C2×D411yescharacteristic
74#1084—22no—
75#1084—41no—
76#1084C22⋊C412yes—
77C2×C3×#41—91no—
78C2×#237—34no—
79C2×C3×M16—91no—
80C2×C3×C4⋊C4—91no—
81C2×ΓL(1,𝔽25)—34no—
82D3×C4⋊C4—34no—
83C3×#103—91no—
84C3×#103—91no—
85C3×#102—34no—
86C3×#102—91no—
87C3×#102C3⋊D311yescharacteristic
88#832—34no—
89#816—34no—
90#810—34no—
91C23×C32—24no—
92C22×C3×D3—34no—
93C32×D4—24no—
94C32×D4C4×D318yescomplemented
95C2×C32×C4—24no—
96C2×C32×C4C22×D314yes—
97C3×#67—616no—
98C2×C3×Dic3—34no—
99C2×C3×Dic3—616no—
100C2×C3×Dic3—64no—
101C2×C3×Dic3—64no—
102C3×C4×D3—38no—
103C3×C4×D3—38no—
104C3×C4×D3—128no—
105C3×Dic6—38no—
106C3×Dic6—64no—
107C32×C8—24no—
108C3×C3⋊C8—64no—
109#417—91noSylow
110C2×H(3,𝔽3)—42no—
111C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
112C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
113C2×H(3,𝔽3)#13711yescharacteristic
114#272—22no—
115C3×#41—94no—
116C3×#41—182no—
117C2×C3×Q8—91no—
118C2×C3×D4—34no—
119C2×C3×D4—181no—
120C2×C3×D4C2×C3⋊D311yescharacteristic
121C22×C3×C4—91no—
122C22×C3×C4—181no—
123C22×C3×C4—181no—
124C2×#67—64no—
125C22×Dic3—64no—
126#237—316no—
127#237—68no—
128C2×C4×D3—34no—
129C2×C4×D3—64no—
130C2×Dic6—34no—
131C3×M16—182no—
132C2×C3×C8—34no—
133C2×C3×C8—91no—
134C2×C3×C8C2×C3⋊D311yescharacteristic
135C3×C4⋊C4—34no—
136C3×C4⋊C4—91no—
137C3×C4⋊C4—181no—
138C3×C4⋊C4C2×C3⋊D311yescharacteristic
139C4⋊Dic3—34no—
140Dic3⋊C4—64no—
141C2×C3⋊C8—34no—
142ΓL(1,𝔽25)—68no—
143C22×C32—28no—
144C22×C32—44no—
145C22×C32—44no—
146C22×C32D3×D414yes—
147C2×C3×D3—38no—
148C2×C3×D3—64no—
149C2×C3×D3—64no—
150C32×C4—44no—
151C32×C4C2×C4×D314yes—
152C32×C4D3×D414yes—
153C3×Dic3—38no—
154C3×Dic3—68no—
155C3×Dic3—124no—
156C2×#41—91no—
157C2×M16—91no—
158C2×C4⋊C4—91no—
159#103—91no—
160#103—91no—
161#102—91no—
162#102#27211yescharacteristic, complemented
163H(3,𝔽3)#41711yescharacteristic, complemented, Sylow
164C23×C3—21no—
165C23×C3—64no—
166C23×C3—91no—
167C22×D3—34no—
168C3×Q8—92no—
169C3×Q8—181no—
170C3×D4—21no—
171C3×D4—38no—
172C3×D4—64no—
173C3×D4—184no—
174C3×D4C4×C3⋊D312yes—
175C2×C3×C4—21no—
176C2×C3×C4—34no—
177C2×C3×C4—64no—
178C2×C3×C4—91no—
179C2×C3×C4—92no—
180C2×C3×C4—92no—
181C2×C3×C4—184no—
182C2×C3×C4—181no—
183C2×C3×C4—181no—
184C2×C3×C4—362no—
185C2×C3×C4C22×C3⋊D311yescharacteristic
186#67—616no—
187C2×Dic3—34no—
188C2×Dic3—616no—
189C2×Dic3—64no—
190C2×Dic3—64no—
191C4×D3—38no—
192C4×D3—38no—
193C4×D3—128no—
194Dic6—38no—
195Dic6—64no—
196C3×C8—21no—
197C3×C8—64no—
198C3×C8—181no—
199C3⋊C8—64no—
200C2×C32—48no—
201C2×C32D3×C22⋊C414yes—
202C2×C32#91114yes—
203C2×C32#91514yes—
204C3×D3—68no—
205#41—94no—
206#41—182no—
207C2×Q8—91no—
208C2×D4—181no—
209C2×D4C2×#27211yescharacteristic
210C22×C4—91no—
211C22×C4—181no—
212C22×C4—181no—
213M16—182no—
214C2×C8—91no—
215C2×C8C2×#27211yescharacteristic
216C4⋊C4—91no—
217C4⋊C4—181no—
218C4⋊C4C2×#27211yescharacteristic
219C22×C3—22no—
220C22×C3—34no—
221C22×C3—41no—
222C22×C3—41no—
223C22×C3—68no—
224C22×C3—92no—
225C22×C3—124no—
226C22×C3—124no—
227C22×C3—181no—
228C22×C3—181no—
229C22×C3D4×C3⋊D311yescharacteristic
230C2×D3—38no—
231C2×D3—64no—
232C2×D3—64no—
233C3×C4—34no—
234C3×C4—34no—
235C3×C4—41no—
236C3×C4—92no—
237C3×C4—124no—
238C3×C4—182no—
239C3×C4—361no—
240C3×C4C2×C4×C3⋊D311yescharacteristic
241C3×C4D4×C3⋊D311yescharacteristic
242Dic3—38no—
243Dic3—68no—
244Dic3—124no—
245C32#477114yescomplemented
246C23—21no—
247C23—91no—
248Q8—92no—
249Q8—181no—
250D4—21no—
251D4—184no—
252D4C4×#27212yescomplemented
253C2×C4—21no—
254C2×C4—91no—
255C2×C4—92no—
256C2×C4—92no—
257C2×C4—184no—
258C2×C4—181no—
259C2×C4—181no—
260C2×C4—362no—
261C2×C4C22×#27211yescharacteristic
262C8—21no—
263C8—181no—
264C2×C3—34no—
265C2×C3—34no—
266C2×C3—34no—
267C2×C3—42no—
268C2×C3—128no—
269C2×C3—182no—
270C2×C3C22⋊C4×C3⋊D311yescharacteristic
271C2×C3#6412011yescharacteristic
272C2×C3#6412411yescharacteristic
273D3—68no—
274C22—22no—
275C22—41no—
276C22—41no—
277C22—92no—
278C22—181no—
279C22—181no—
280C22D4×#27211yescharacteristic
281C4—41no—
282C4—92no—
283C4—182no—
284C4—361no—
285C4C2×C4×#27211yescharacteristic
286C4D4×#27211yescharacteristic
287C3—34no—
288C3#1059247011yescharacteristic
289C2—42no—
290C2—182no—
291C2C22⋊C4×#27211yescharacteristic
292C2#1172179311yescharacteristic
293C2#1172180111yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#102C2
prime radicalC2×#88744C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×#102C3⋊D3
hypocenterH(3,𝔽3)#417
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#88527C2semidirect
3. C4⋊C4×#272C2semidirect
4. H(3,𝔽3)×#102C2semidirect
5. #11720578C2semidirect
6. #11720549C2semidirect
7. #11720532C2semidirect
13. #88744C4semidirect
26. C32×#102D3semidirect
33. D4×H(3,𝔽3)C2×C4semidirect
45. #6229C2×C4semidirect
94. C32×D4C4×D3semidirect
162. #102#272semidirect
163. H(3,𝔽3)#417semidirect
245. C32#4771semidirect
252. D4C4×#272semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#887442
C2×#885272
C4⋊C4×#2722
H(3,𝔽3)×#1022
#117205782
#117205492
#117205322
C3×#47713